We remark that very recently Allen, Böttcher, Corsten, Davies, Jenssen, Morris, Roberts, and Skokan [1]} proved (ii) for \(r=3\) , but without spread techniques and using techniques related to Johansson–Kahn–Vu [2]}. Results falling under (i) have been independently proved by Kang, Kelly, Kühn, Osthus, and Pfenninger [3]} using a similar strategy but with differences in implementation. Additionally, for the Dirac threshold for \((k-1)\) -degrees, they prove a sharper “stability” version; we refer to the discussion after thm:pm for more details.
